Pochhammer symbol

In mathematics, the Pochhammer symbol introduced by Leo August Pochhammer is the notation (x)n, where n is a non-negative integer. Depending on the context the Pochhammer symbol may represent either the rising factorial or the falling factorial as defined below. Care needs to be taken to check which interpretation is being used in any particular article. Pochhammer himself actually used (x)n with yet another meaning, namely to denote the binomial coefficient.[1]

In this article the Pochhammer symbol (x)n is used to represent the falling factorial (sometimes called the "descending factorial",[2]"falling sequential product", "lower factorial"):

In this article the symbol x(n) is used for the rising factorial (sometimes called the "Pochhammer function", "Pochhammer polynomial", "ascending factorial",[2]"rising sequential product" or "upper factorial"):

These conventions are used in combinatorics.[3] However in the theory of special functions (in particular the hypergeometric function) the Pochhammer symbol (x)n is used to represent the rising factorial.[4] A useful list of formulas for manipulating the rising factorial in this last notation is given in (Slater 1966, Appendix I). Knuth uses the term factorial powers to comprise rising and falling factorials.[5]

When x is a non-negative integer, then (x)n gives the number of n-permutations of an x-element set, or equivalently the number of injective functions from a set of size n to a set of size x. However, for these meanings other notations like xPn and P(x,n) are commonly used. The Pochhammer symbol serves mostly for more algebraic uses, for instance when x is an indeterminate, in which case (x)n designates a particular polynomial of degree n in x.

The falling factorial occurs in a formula which represents polynomials using the forward difference operatorΔ and which is formally similar to Taylor's theorem of calculus. In this formula and in many other places, the falling factorial (x)k in the calculus of finite differences plays the role of xk in differential calculus. Note for instance the similarity of

Since the falling factorials are a basis for the polynomial ring, we can re-express the product of two of them as a linear combination of falling factorials:

The coefficients of the (x)m+n−k, called connection coefficients, have a combinatorial interpretation as the number of ways to identify (or glue together) k elements each from a set of size m and a set of size n.

goes back to A. Capelli (1893) and L. Toscano (1939), respectively.[6] Graham, Knuth and Patashnik[7] propose to pronounce these expressions as "x to the m rising" and "x to the m falling", respectively.

Other notations for the falling factorial include P(x, n), xPn, Px,n, or xPn. (See permutation and combination.)

An alternate notation for the rising factorial x(n) is the less common (x)+n. When the notation (x)+n is used for the rising factorial, the notation (x)–n is typically used for the ordinary falling factorial to avoid confusion.[1]